Hexapod kinematic mountings for optical elements, and optical systems comprising same

ABSTRACT

“Hexapod” mountings are disclosed for use with optical elements. An exemplary mounting includes a base, a platform that is movable relative to the base, and six legs having nominally identical length. Three pairs of legs, having substantially equal stiffness, extend between the base and platform and support the platform relative to the base. In each pair of legs, respective first ends are coupled together in a Λ-shaped manner forming a respective apex. Respective second ends are splayed relative to the apex, desirably forming an angle of substantially 109.5° at the apex. The apices are mounted equidistantly from each other on a circle on the platform. The respective second ends of the pairs of legs are mounted at respective locations on a circle on the base. The axes of each pair of legs define a respective leg plane substantially perpendicular to the base plane. Each leg has an actuator that, when energized, changes a length of the respective leg. Coordinated energization of the actuators in selected legs produces a desired movement of the platform relative to the base in all six degrees of freedom of motion.

CROSS REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application No.60/813,481, filed Jun. 13, 2006, incorporated herein by reference in itsentirety.

FIELD

This disclosure pertains to, inter alia, adjustable mounting structuresfor optical elements, to assemblies of optical elements comprising atleast one optical element mounted on an adjustable mounting structure,and to optical systems comprising one or more such assemblies. Morespecifically, the mounting structure includes a “Stewart Platform,”which is a platform mounted on six legs. The mounting structures andassemblies are especially advantageous when used for mounting mirrors ina manner providing active mirror adjustment in multiple degrees offreedom of motion. An advantageous application of such an assembly is ina projection-optical system as used for performing microlithography,such as extreme ultraviolet (EUV) lithography (EUVL).

BACKGROUND

Imaging technology in many fields has advanced greatly in recent years,aided especially by substantial advancements in the art and science ofoptical design and by the debut of “adaptive optics,” in which real-timechanges can be deliberately made to an optical property of an individualoptical element. Consequently, various optical imaging systems havebecome available that exhibit performance levels considered impossibleonly a short time ago. One field in which such advancements have beenachieved is astronomy, in which sophisticated telescopes have recentlybeen developed that produce exceptionally good resolution ofastronomical objects. Another field is terrestrial imaging from space,in which sophisticated and powerful imaging satellites provide extremelywell-resolved images of the surface of the earth or other celestialbody. Yet another field is microlithography, a technology used forimaging microcircuit patterns and the like on the surfaces of siliconwafers or other substrates.

Since the beginning of microlithography, so-called “opticalmicrolithography” (projection lithography performed using deepultraviolet light transmitted through optical systems that are at leastpartially refractive) has been the workhorse imaging technology forforming microcircuits on silicon wafers and other substrates. Ascritical dimensions of microcircuits have become increasinglyminiaturized (currently less than 100 nm), accompanied by prodigiousincreases in the density of active circuit elements being formed inmicrocircuits, the resolution limitations of optical microlithographyhave become apparent. This has fueled an urgent need for a practical“next-generation” lithography (NGL) capable of resolving smaller patternfeatures than can be imaged using optical microlithography.

Extreme-ultraviolet lithography (EUVL) is currently regarded as a viablecandidate next-generation lithography offering good prospects ofsubstantially finer pattern resolution than currently obtainable usingconventional optical lithography. The expectations of increasedresolution from EUVL stem largely from the fact that, whereas currentoptical lithography is performed using an imaging wavelength in therange of 150-250 nm, EUVL is performed using an imaging wavelength inthe range of 11-5 nm, which is at least ten times shorter than theshortest conventional “optical” wavelengths. Generally, the shorter thewavelength of light being used for pattern imaging in microlithography,the finer the obtainable resolution.

In view of the extremely small pattern elements (critical dimensions ofless than 70 nm) that can be resolved using EUVL, the accuracy andprecision with which pattern imaging is performed using this technologymust be extremely high to ensure proper resolution, placement, andregistration of multiple pattern layers on a substrate and to ensurethat the pattern elements are transferred to the substrate with highfidelity. To obtain such high accuracy and precision, extreme effortsare being expended in the design and configuration of optical systemsused in EUVL systems.

EUVL differs substantially from optical microlithography in that noknown materials are sufficiently transmissive and refractive to EUVlight to be useful for making EUV lenses. Consequently, EUVL opticalsystems consist of reflective optical elements, namely EUV-reflectivemirrors. In one type of conventional EUVL system currently underdevelopment, a reflective “illumination-optical system” (comprisingmultiple mirrors) is used for illuminating a pattern-defining reticlewith a beam of EUV light, and a reflective “projection-optical system”(also comprising multiple mirrors) is used for projecting an imagingbeam from the reticle to the wafer or other substrate. Most, if not all,of the mirrors in the illumination-optical system and projection-opticalsystem are “multilayer-film” mirrors, which are the only known types ofmirrors (besides grazing-incidence mirrors) that exhibit useful levelsof reflectivity to incident EUV light.

A conventional EUVL system 100 is shown schematically in FIG. 7. Thedepicted system 100 comprises a laser-plasma EUV source ES, anillumination-optical system IL comprising five mirrors IM₁-IM₅, and aprojection-optical system PL comprising six mirrors PM₁-PM₆. The sourceES comprises a high-power pulsed laser 111 and a convex lens 112 thatconverges the beam produced by the laser 111 to a point 113. The sourceES also includes a conduit 114 for a suitable target material that isdischarged at the point 113 where the incident laser beam converts thetarget material to a plasma. The plasma generates various wavelengths oflight, including EUV light, that is “collected” by an elliptical mirror115 and delivered convergently to the illumination-optical system IL. Avacuum conduit 116 removes debris produced by the plasma.

The illumination-optical system IL is situated between the source ES anda pattern-defining reflective reticle M, and the projection-opticalsystem PL is situated optically between the reticle and a wafer W orother substrate onto which the projection-optical system projects thepattern. Although the depicted projection-optical system PL comprisessix multilayer-film mirrors PM₁-PM₆, other configurations for EUVLprojection-optical systems have four mirrors or as few as two mirrors.The mirrors are arranged so as to conserve space between the reticle Mand wafer W, with the reticle and wafer being located on opposing endsof the projection-optical system PL to allow sufficient room for areticle stage RS and a wafer stage WS and to satisfy other practicalconsiderations.

To achieve high imaging resolution in EUVL, extreme demands are imposedon the configuration and operation of the optical systems, especiallythe projection-optical system PL. For example, the reflective surfacesof the mirrors (e.g., PM₁-PM₆) must have extremely low figure errors(e.g., <0.25 nm), the surficial multilayer films must be formed withextremely high accuracy and precision, and the mirrors must be mountedwith extremely high positional accuracy and stability in a rigid frameor “barrel” that does not deform and that isolates the mirrors fromexternal vibrations. Experience has shown that even these measures donot produce optimal optical performance, especially over time and/orfrom one optical system to the next. For example, significant variationsin imaging performance can exist from one exposure die to the next onthe surface of the wafer. In addition, the mirrors and their supportingstructures can experience thermal deformations, vibrations, and otherstresses that change with time. Also, even though movements of thereticle stage RS and wafer stage WS are synchronized and independentlycontrolled to achieve very accurate positioning of these components fromdie to die of exposure, these controls are insufficient for achievingthe level of optical performance currently being demanded from EUVLsystems.

To improve optical performance of EUVL systems further, adaptive-opticalschemes such as active mirror adjustment (AMA) schemes are beingconsidered. In one type of AMA scheme one or more of the mirrors(especially of the projection-optical system PL) are mounted in a mannerallowing positional adjustments of the respective mirror(s) to be madeduring actual use of the optical system. Certain projection-opticalsystems currently in use employ, for at least one mirror of the system,a mounting that provides three degrees of freedom (DOF) of motion.Whereas such adjustability provides some benefits, it falls short of theideal full six DOF (x, y, z, θ_(x), θ_(y), θ_(z)) of adjustability.

A type of kinematic mounting that has found some utility, particularlyin certain types of astronomical telescopes (notably having acatadioptric configuration), is the so-called “hexapod” mountingconventionally known as a “Stewart Platform” or “Stewart-GoughPlatform.” Stewart platforms also are being used for mounting mirrorsand other optical elements used in space-borne optical systems. As usedherein, “hexapod” refers to a mounting structure by which a platform issupported relative to a base by six legs, in the general manner of aStewart Platform or Stewart-Gough Platform, but does not encompassvarious six-legged, self-propelled robotic devices (also referred to inthe literature as “hexapods”) that walk about or otherwise perambulateusing their legs in a walking-insect manner.

A conventional hexapod mount (Stewart Platform) 120 is depicted in FIG.8. A mirror 122 (as an exemplary payload) is mounted to a platform 124by multiple holds 126. The platform 124 is supported relative to astationary base 128 by three pairs of legs 130. The legs 130 of eachpair have upper ends 132 that are mounted via respective sphericalbearings 134 to the underside of the platform 124 where the upper endsof the legs converge. The spherical bearings 134 allow changes in theangles of the legs 130 relative to the platform 124. The sphericalbearings 134 are equally spaced from one another on the underside of theplatform 124. From the platform 124 to the base 128 the legs 130 of eachpair are divergent (splayed), and the bottom ends 138 of the legs areattached to the base by respective spherical bearings 140 that allowchanges in the angles of the legs relative to the base. Each leg 130includes a respective actuator (not detailed) that is operable to changethe length of the leg. Each actuator is independently operable, and theactuators typically are operated in a coordinated manner to achievemotions of the platform 124 (and hence of the mirror 122) relative tothe base 128 in all six DOF. Thus, whereas each leg 130 when actuatedexhibits only one DOF of translational motion (by increasing ordecreasing in length), coordinated actuations of the legs provide thefull six DOF of movement of the platform 124 and mirror 122 relative tothe base 128.

Various types of leg actuators are used in conventional Stewartplatforms. One type of actuator is a hydraulic cylinder. Another type isa motor connected to a rotatable screw threaded into a bearing such as aball-screw. Yet other types are linear servo motors and voice-coilmotors.

Whereas a conventional hexapod mount as summarized above is satisfactoryfor certain applications (such as conventional astronomical telescopesand other systems requiring accuracy and precision in the micrometerrange), it is unsatisfactory for use in EUVL systems and other opticalsystems requiring accuracy and precision in the nanometer range (i.e.,three or more orders of magnitude greater accuracy and precision than inconventional systems). In other words, conventional hexapod mounts havecertain limitations that have prevented them from being usedsatisfactorily in EUVL systems and other demanding applications.

One limitation of many types of conventional hexapods is their use ofspherical bearings for coupling the upper and lower ends of the legs tothe platform and base, respectively. Spherical bearings inherentlyexhibit friction and backlash that usually do not cause significantproblems whenever the required accuracy and precision of the hexapod isin the micrometer range, but are intolerable whenever the requiredaccuracy and precision are in the nanometer range or less. Sphericalbearings also tend to transmit vibrations, to and from the platform (andoptical element supported thereby), that degrade optical performance.

Another limitation of many types of conventional hexapods is theparticular types of actuators used for increasing or decreasing thelengths of the legs. Conventional hexapods that support large masses(such as large telescope mirrors) use hydraulic cylinders or linearmotors for actuating the legs. Smaller conventional hexapods generallyuse micromotors, servomotors, or voice-coil motors. These types ofactuators are too imprecise for use in hexapods for EUVL systems andother systems demanding precision in the nanometer range.

Yet another limitation of conventional hexapods is their tendency toexhibit “cross-coupling.” Cross-coupling is an unintended change in oneor more movement parameters (x, y, z, θ_(x), θ_(y), θ_(z)) accompanyinga change in another movement parameter. One example is an unwantedy-direction and/or z-direction shift accompanying actuation of thehexapod to achieve a change in x-direction position. Another example isan unwanted change in θ_(x), θ_(y), and/or θ_(z) accompanying actuationof the hexapod to achieve a change in x-, y-, and/or z-directionposition. Whereas significant cross-coupling can be offset usingfeedback control of the platform position, achieving the offsettingactuations of the legs requires massive control software and consumestime that would substantially decrease throughput of an EUVL system.

Therefore, there is a need for improved hexapod mountings, useful formounting mirrors and other types of optical elements for activeadjustment, that provide an operational precision in the nanometer rangeand that exhibit substantially no cross-coupling errors.

SUMMARY

The needs articulated above, as well as other needs, are satisfied byhexapod mountings, and optical systems comprising same, as disclosedherein.

A first aspect is directed to “hexapod” mountings. These mountings aregenerally configured in the manner generally known as “Stewart” or“Stewart-Gough” platforms, but have certain distinctive featurescompared to conventional platforms. An embodiment of such a mountingcomprises a base, a platform, and six legs. The base defines a baseplane. The platform is situated relative to the base and is movablerelative to the base. The legs each have nominally identical length anda respective leg axis. The legs have substantially equal stiffness andare arranged in three pairs that extend between the base and platformand that support the platform relative to the base. Each pair of legshas first and second ends. The first ends of each pair are coupledtogether in a Λ-shaped manner forming a respective apex. The respectivesecond ends are splayed relative to the apex The apices are mountedequidistantly from each other at respective locations on a circle on theplatform. The respective second ends of the pairs of legs are mounted atrespective locations on a circle on the base such that the respectiveaxes of each pair of legs define a respective leg plane that issubstantially perpendicular to the base plane. Each leg comprises anactuator serving, when energized, to change a length of the respectiveleg such that a coordinated energization of the respective actuators inselected legs produces a desired movement of the platform relative tothe base in all six degrees of freedom of motion (namely, x, y, z,θ_(x), θ_(y), θ_(z)).

Desirably, the respective legs of each pair form an angle ofsubstantially 109.5° at the apex.

The leg actuators desirably are respective piezoelectric actuators, butalternatively can be any of various other types of actuators such asvoice-coil motor, pneumatic, linear motor, etc., or combinations ofthese. Each actuator desirably comprises a respective coarse actuatorand a respective fine actuator, a configuration that is especiallyamenable to piezoelectric actuators, and can be used with other types ofactuators. The coarse actuator and the fine actuator desirably arearranged in tandem along the respective leg axis, or in any othersuitable arrangement.

In one embodiment each leg comprises at least one respectiveleg-extension flexure(s) situated relative to the leg actuator toprovide at least one, but not all six, degrees of freedom of motionaccompanying leg extension and retraction caused by the respective legactuator. Each leg also desirably comprises a respective leg-lengthmonitor. The leg-length monitors are useful in, for example, a feed-backsystem for ensuring that the platform moves in a desired manner andamount accompanying extension or retraction of one or more of the legs.

In another embodiment the first ends of each pair of legs compriserespective flexures providing the respective end with at least two, butnot all six, degrees of freedom of motion. Also, the second ends of eachpair of legs comprise respective flexures providing the respective endwith at least two, but not all six, degrees of freedom of motion. Eachleg also desirably comprises a respective leg-length monitor.

Any of the embodiments summarized above can further comprise at leastone height monitor that is situated and configured to measure andmonitor position (height) of the platform relative to a fixed reference.The fixed reference can be the base. The height monitor is useful in,for example, a feed-back system for ensuring that the platform moves ina desired manner and amount accompanying extension or retraction of oneor more of the legs.

According to another aspect, kinematically mounted optical elements areprovided. An embodiment of the same comprises an optical element, abase, a platform, at least one hold affixing the optical element to theplatform, and a hexapod situated between the base and the platform. Thebase defines a base plane, and the platform is movable relative to thebase. The hexapod supports the platform relative to the base andcomprises six legs having nominally identical length. Each leg has arespective leg axis. The legs have substantially equal stiffness and arearranged in three pairs each having first and second ends. The firstends of each pair are coupled together in a Λ-shaped manner forming arespective apex. (Desirably, the respective legs of each pair form anangle of substantially 109.5° at the apex.) The respective second endsare splayed relative to the apex. The apices are mounted equidistantlyfrom each other at respective locations on a circle on the platform. Therespective second ends of the pairs of legs are mounted at respectivelocations on a circle on the base such that the respective axes of eachpair of legs define a respective leg plane that is substantiallyperpendicular to the base plane. Each leg comprises an actuator thatserves, when energized, to change a length of the respective leg suchthat a coordinated energization of the respective actuators in selectedlegs produces a desired movement of the platform relative to the base inall six degrees of freedom of motion.

The optical element can be any of various types and configurations, suchas (but not limited to) a mirror or mirror group, a lens or lens group,a filter or filter group, or other type of optical element(s), orcombinations of the same.

According to another aspect, optical systems are provided. An embodimentof such a system comprises a frame, a base, a platform, at least oneoptical element mounted to the platform, and a hexapod situated betweenthe base and the platform. The base is mounted to the frame and definesa base plane, and the platform is movable relative to the base. The atleast one optical element is mounted to the platform. The hexapod issituated between the base and the platform so as to support the platformrelative to the base. The hexapod comprises six legs having nominallyidentical length and substantially equal stiffness. Each leg has arespective leg axis, and the legs are arranged in three pairs eachhaving first and second ends. The first ends of each pair are coupledtogether in a Λ-shaped manner forming a respective apex, and therespective second ends are splayed relative to the apex. Desirably, therespective legs of each pair form an angle of substantially 109.5° atthe apex. The apices are mounted equidistantly from each other atrespective locations on a circle on the platform. The respective secondends of the pairs of legs are mounted at respective locations on acircle on the base such that the respective axes of each pair of legsdefine a respective leg plane that is substantially perpendicular to thebase plane. Each leg comprises an actuator that serves, when energized,to change a length of the respective leg such that a coordinatedenergization of the respective actuators in selected legs produces adesired movement of the platform relative to the base in all six degreesof freedom of motion.

The leg actuators desirably are respective piezoelectric actuators, butalternatively can be any of various other types of piezoelectricactuators, as noted above. Each piezoelectric actuator can comprise arespective coarse actuator and a respective fine actuator, as noteabove.

Each leg can further comprise at least one respective leg-extensionflexure(s) situated relative to the leg actuator to provide at leastone, but not all six, DOF of motion accompanying leg extension andretraction caused by the respective leg actuator. As noted above, eachleg can further comprise a respective leg-length monitor.

In another embodiment the first ends of each pair of legs compriserespective flexures that provide the respective end with at least two,but not all six, DOF of motion. Similarly, the second ends of each pairof legs comprise respective flexures providing the respective end withat least two, but not all six, DOF of motion.

Various embodiments of an optical system as summarized above can furthercomprise at least one monitor situated and configured to measure andmonitor position of the optical element relative to a fixed reference.

The optical element can be a reflective optical element or any ofvarious other types of optical elements, as noted above. A particularlyadvantageous optical system incorporating reflective optical elements isan EUVL optical system, such as (but not limited to) an EUVLprojection-optical system.

The optical system can be one that comprises multiple optical elements.In such a system, at least one optical element can be mounted to theframe by a respective base, platform, and hexapod.

Yet another aspect is directed to kinematically mounted opticalelements. An embodiment of the same comprises an optical element, abase, a platform, at least one hold affixing the optical element to theplatform, and a hexapod. The base defines a base plane, and the platformis movable (via the hexapod) relative to the base. The hexapod issituated between the base and the platform so as to support the platformrelative to the base. The hexapod comprises six legs having nominallyidentical length. Each leg has a respective leg axis, and the legs arearranged in three pairs each having first and second ends. The firstends of each pair are coupled together in a Λ-shaped manner forming arespective apex, and the respective second ends are splayed relative tothe apex. The apices are mounted equidistantly from each other atrespective locations on a circle on the platform. The respective secondends of the pairs of legs are mounted at respective locations on acircle on the base such that the respective axes of each pair of legsdefine a respective leg plane that is substantially perpendicular to thebase plane. Each leg comprises an actuator that serves, when energized,to change a length of the respective leg such that a coordinatedenergization of the respective actuators in selected legs produces adesired movement of the platform, with substantially no cross-coupling,relative to the base in all six degrees of freedom of motion.

Desirably, the legs have substantially equal thickness. Furtherdesirably, the legs of each pair form an angle of substantially 109.5°at the apex.

According to another aspect, an optical system is provided thatcomprises an optical element such as that summarized above in thepreceding paragraph.

The foregoing and additional features and advantages of the inventionwill be more readily apparent from the following detailed description,which proceeds with reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of certain relationships of an embodimentof a hexapod mounting, showing the locations at which the legs areattached to base and platform and the two coordinate axes of the system.

FIG. 2 is a plan view of FIG. 1, showing locations at which legs areattached and the relationships of those locations.

FIG. 3(A) is a plan schematic diagram showing certain geometricrelationships of the points at which the legs are attached, and thedesignations of key geometric variables.

FIG. 3(B) is a vertical schematic diagram (orthogonal to FIG. 3(A))showing certain other geometric relationships and variable designations.

FIG. 4(A) is a perspective view of a hexapod mounting according to arepresentative embodiment.

FIG. 4(B) is a plan view of hexapod mounting shown in FIG. 4(A).

FIG. 5 is a perspective view of a pair of legs used in the embodiment ofFIGS. 4(A) and 4(B).

FIG. 6 is a schematic diagram of a projection-optical system includingone optical element on a hexapod mounting as described herein.

FIG. 7 is a schematic optical diagram of a conventionalextreme-ultraviolet (EUV) lithography system, including EUV source,illumination-optical system, and projection-optical system.

FIG. 8 is a perspective view showing the arrangement of legs of aconventional hexapod configured as a Stewart platform.

DETAILED DESCRIPTION

This disclosure is set forth in the context of a representativeembodiment that is not intended to be limiting in any way. Also, eventhough the embodiment is described in the context of holding an opticalmirror, it will be understood that the mirror alternatively can beanother type of optical element or other object. Hence, an“active-mirror-adjustment” (AMA) mechanism as described below is notlimited to use with a mirror. In addition, positional terms such as“above,” below,” “upper,” “lower,” “over,” “under,” “horizontal,” and“vertical” are used to facilitate comprehension of spatialrelationships, but are not intended to be limited to their literalmeanings in the context of a terrestrial environment.

The particular AMA mechanism that is the subject of this disclosure is aso-called “hexapod” mount configured as a Stewart platform. At least oneoptical element of an EUVL system or other optical system is mounted onthe platform, which is mounted by six legs to a base and is movablerelative to the base in six degrees of freedom (DOF). Movement of theplatform is feedback-controlled in a manner that provides a highbandwidth (very rapid response time) and extremely high accuracy andprecision, which are advantageous for use in an EUVL system. Another keyadvantage of the instant AMA mechanism is its exhibited absence ofcoupling effects (zero coupling stiffness) and high axial stiffness.Both these characteristics are discussed in more detail later.

A representative embodiment of an AMA mechanism is depicted in FIGS. 1and 2. The mechanism comprises a platform 12, a base 14, and six legsL₁-L₆ arranged into three pairs L₁ and L₂, L₃ and L₄, and L₅ and L₆. Theposition of the base is typically fixed, achieved by mounting the baseto a rigid frame, in an optical “barrel” or “column,” or analogousstructure. The legs L₁-L₆ support the platform 12 relative to the base14 in a manner allowing movement of the platform relative to the base.The “upper” ends of each pair of legs L₁-L₂, L₃-L₄, L₅-L₆ converge at arespective apex attached at a respective location B₁₂, B₃₄, B₅₆ on theundersurface of the platform 12, and the “lower” ends of each pair oflegs are attached to the upper surface of the base 14 at respectivelocations A₁ and A₂, A₃ and A₄, A₅ and A₆. The locations B₁₂, B₃₄, B₅₆are located equally spaced from each other on a circle 16 (on theunderside of the platform 12) having a center at O_(B), and thelocations A₁-A₆ are located on a circle 18 (on the upper side of thebase 14) having a center at O_(A).

In a “null” condition as shown, all the legs L₁-L₆ nominally haveidentical length, the platform 12 is exactly parallel to the base 14,the moving coordinates (u, v, w) of the platform are coincident withfixed coordinates (x, y, z) of the base, and the center O_(A) of thebase and center O_(B) of the platform are on a vertical axis A_(x).Also, the “vertical” axis w of the moving coordinates and the “vertical”axis z of the fixed coordinates are on the vertical axis A_(x), and thecentroid of the platform 12 is at an elevation h= O_(A)O_(B) above thebase 14. Starting from the null condition, a change in length of any oneor more of the legs L₁-L₆ causes a shift of the moving coordinates(u,v,w) relative to the fixed coordinates (x,y,z). The shift can reflecta respective change in one or more degrees of freedom (x, y, z, θ_(x),θ_(y), θ_(z)) of motion of the platform 12 relative to the base 14.

In the following discussion, a, b, c, and L are design parameters thatare defined as follows (see FIGS. 3(A) and 3(B)):

A_(i) is any of A₁-A₆,

B_(i) is any of B₁-B₆,

B_(ij) is any of B₁₂, B₃₄, B₅₆,

a is the radius from O_(A) to any point A_(i) on the base: a= O_(A)A_(i),

b is the radius from O_(B) to any point B_(i) on the platform: b=O_(B)B_(i) ,

c is the “leg interval”, wherein 2c= A₁A₂ = A₃A₄ = A₅A₆ , and

L is the length of a leg.

The points A_(i) and B_(ij) have respective coordinates (e.g., x, y, z),as follows:

A ₁ =[−c,√{square root over (a ² −c ²)},0]  (1)

A ₂ =[c,√{square root over (a ² −c ²)},0]  (2)

$\begin{matrix}{A_{3} = \left\lbrack {{\frac{c}{2} + {\frac{\sqrt{3}}{2}\sqrt{a^{2} - c^{2}}}},{{\frac{\sqrt{3}}{2}c} - {\frac{1}{2}\sqrt{a^{2} - c^{2}}}},0} \right\rbrack} & (3) \\{A_{4} = \left\lbrack {{\frac{- c}{2} + {\frac{\sqrt{3}}{2}\sqrt{a^{2} - c^{2}}}},{{\frac{- \sqrt{3}}{2}c} - {\frac{1}{2}\sqrt{a^{2} - c^{2}}}},0} \right\rbrack} & (4) \\{A_{5} = \left\lbrack {{\frac{c}{2} - {\frac{\sqrt{3}}{2}\sqrt{a^{2} - c^{2}}}},{{\frac{- \sqrt{3}}{2}c} - {\frac{1}{2}\sqrt{a^{2} - c^{2}}}},0} \right\rbrack} & (5) \\{A_{6} = \left\lbrack {{\frac{- c}{2} - {\frac{\sqrt{3}}{2}\sqrt{a^{2} - c^{2}}}},{{\frac{\sqrt{3}}{2}c} - {\frac{1}{2}\sqrt{a^{2} - c^{2}}}},0} \right\rbrack} & (6)\end{matrix}$

B₁₂=[o,b,h]  (7)

$\begin{matrix}{B_{34} = \left\lbrack {{\frac{- \sqrt{3}}{2}b},{\frac{- 1}{2}b},h} \right\rbrack} & (8) \\{B_{56} = \left\lbrack {{\frac{- \sqrt{3}}{2}b},{\frac{- 1}{2}b},h} \right\rbrack} & (9)\end{matrix}$

To determine h, consider that L= A₁B₁₂ . If A₁=[−c, √{square root over(a²−c²)}, 0] and B₁₂=[0, b, h], then L²= A₁B₁₂ ²=c²+(b−√{square rootover (a²−c²)})²+h². Solving for h² yields:

h ² =L ² −b ²+2b√{square root over (a ² −c ²)}−a ².  (10)

Exemplary coordinates are as follows (dimensions are in mm):

Point x y z A₁ 75.6329 −179.0000 20.0000 A₂ −75.6329 −179.0000 20.0000A₃ −192.8350 24.0000 20.0000 A₄ −117.2021 155.0000 20.0000 A₅ 117.2021155.0000 20.0000 A₆ 192.8350 24.0000 20.0000 B₁₂ 0.0000 −179.0000151.0000 B₃₄ −155.0185 89.5000 151.0000 B₅₆ 155.0185 89.5000 151.0000From these coordinates, the following exemplary values can be obtained:

-   -   a=((−179)²+(75.6329)²)^(1/2)=194.32 (from coordinate A₁)    -   b=179 (from coordinate B₁₂)    -   c=75.6329 (from coordinate A₁ or A₂)

The vector d_(i) is a corresponding elongation vector for the respectiveleg. Since there are six legs, i=1, 2, 3, . . . , 6. In other words,d₁={right arrow over (A₁B₁₂)}, d₂={right arrow over (A₂B₁₂)}, d₃={rightarrow over (A₃B₃₄)}, d₄={right arrow over (A₄B₃₄)}, d₅={right arrow over(A₅B₅₆)}, and d₆={right arrow over (A₆B₅₆)}. The elongation vectors canbe collectively denoted by the vector q=[d₁, d₂, d₃, d₄, d₅, d₆]^(T). Ifthe position of the platform is denoted by the vector x, the kinematicalconstraints imposed by the legs can be expressed in the general form:

f(x,q)=0  (1)

Differentiating this expression with respect to time yields arelationship between leg-elongation velocity ({dot over (q)}) and anoutput-velocity vector ({dot over (x)}) for the platform:

J_(x){dot over (x)}=J_(q){dot over (q)}  (12)

where

$J_{x} = \frac{\partial f}{\partial x}$

and

$J_{q} = {- {\frac{\partial f}{\partial q}.}}$

The derivation leads to two separate Jacobian matrices. The overallJacobian matrix, J, can be written:

{dot over (q)}=J{dot over (x)},  (13)

thus, J=J_(q) ⁻¹J_(x). Note that, in general, the Jacobian matrix mapsoutput velocities (leg-joint velocities) to leg-elongation velocities.The output-velocity vector {dot over (x)} can be described by thevelocity (v_(P)) of the centroid P and the angular velocity (ω_(B)) ofthe platform, thus:

$\begin{matrix}{\overset{.}{x} = {\begin{bmatrix}v_{P} \\\omega_{B}\end{bmatrix}.}} & (14)\end{matrix}$

A loop-closure equation for each leg can be written as:

OP+ PB _(i) = OA _(i)+ A _(i) B _(i)  (15)

Differentiating this equation with respect to time yields:

v _(P)+ω_(B) ×b _(i) =d _(i)ω_(i) ×s _(i) +d _(i) s _(i)  (16)

where b_(i) denotes the vector {right arrow over (PB)}_(i), s_(i) is theunit vector along {right arrow over (A_(i)B)}_(i) (i.e.,

$s_{i} = \frac{\overset{\rightarrow}{A_{i}B_{i}}}{\overset{\rightarrow}{A_{i}B_{i}}}$

), and ω_(i) is the angular velocity of the i^(th) leg with respect tothe fixed reference frame A. To eliminate ω_(i), both sides of equation(16) are dot-multiplied by s_(i):

s _(i) ·v _(p)+(b _(i) ×s _(i))·ω_(b) ={dot over (d)} _(i)  (17)

Rewriting equation (17) for each leg yields J_(x)(x_(i))=J_(q)(q_(i)),where J_(x)=[s_(i) ^(T)(b_(i)×s_(i))^(T)] and J_(q)=I. The kinematicJacobian then can be computed using the relation J=J_(q) ⁻¹J_(x).

Equation (17) can be assembled as equation (12) with the vector {dotover (x)} (equation (14)), where:

$\begin{matrix}{J_{x} = \begin{bmatrix}s_{1}^{T} & \left( {b_{1} \times s_{1}} \right)^{T} \\\cdots & \cdots \\s_{6}^{T} & \left( {b_{6} \times s_{6}} \right)^{T}\end{bmatrix}} & (18)\end{matrix}$

and

J _(q) =I(a 6×6 identity matrix)  (19)

Note again that J=J_(q) ⁻¹J_(x).

Based on the principle of virtual work, at equilibrium:

δW=τ ^(T) δq−F ^(T) δx=0  (20)

where F=[f, n]′ is the applied force (a vector) to the platform to movethe platform, and τ=[τ₁, τ₂, . . . , τ₆]′ (or [f₁, f₂, . . . , f₆]′)represents the vector force applied by the actuated legs.

Equation (13) can be written as a virtual displacement, or kinematicJacobian, relationship:

δq=Jδx  (21)

wherein δq is incremental leg movement, which for small changes can beexpressed:

$\begin{matrix}{{\Delta \begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4} \\q_{5} \\q_{6}\end{bmatrix}} = {J\; {\Delta \begin{bmatrix}x \\y \\z \\\theta_{x} \\\theta_{y} \\\theta_{z}\end{bmatrix}}}} & (22)\end{matrix}$

and δx is incremental displacement of the platform. From equations (20)and (21) can be obtained:

F=J^(T)τ  (23)

where J, from the relationship J=J_(q) ⁻¹J_(x) and from equation (18),is as follows:

$\begin{matrix}{J = {{J_{q}^{- 1}J_{x}} = \begin{bmatrix}s_{1}^{T} & \left( {b_{1} \times s_{1}} \right)^{T} \\\cdots & \cdots \\s_{6}^{T} & \left( {b_{6} \times s_{6}} \right)^{T}\end{bmatrix}}} & (24)\end{matrix}$

where b_(i) denotes the vector {right arrow over (PB)}_(i), and s_(i) isa unit vector as defined above.

Equation (23) becomes:

$\begin{matrix}{\begin{bmatrix}f \\n\end{bmatrix} = {\begin{bmatrix}s_{1} & s_{2} & \ldots & s_{6} \\{b_{1} \times s_{1}} & {b_{2} \times s_{2}} & \cdots & {b_{6} \times s_{6}}\end{bmatrix}\begin{bmatrix}f_{1} \\f_{2} \\\vdots \\f_{6}\end{bmatrix}}} & (25)\end{matrix}$

where f represents translational forces, n denotes the moment torques,and f_(i) is the respective force generated by an actuated leg.

Equation (25) can also be obtained in the following manner. The forceacting on the moving platform by each leg can also be written as:

f_(i)=f_(i)S_(i),  (26)

for i=1, 2, . . . , 6, and S_(i)=d_(i)/d_(i), defined as above. Summingall the forces acting on the moving platform,

$\begin{matrix}{{\sum\limits_{i = 1}^{6}\; {f_{i}S_{i}}} = f} & (27)\end{matrix}$

and summing the moments contributed by all forces about the centroid Pof the moving platform yields:

$\begin{matrix}{{\sum\limits_{i = 1}^{6}\; {f_{i}b_{i} \times S_{i}}} = n} & (28)\end{matrix}$

Equation (25) is obtained by combining equations (26) and (27).

The vector force τ applied by actuated legs can be defined as:

τ=χΔq  (29)

where χ=diag [k₁, k₂, . . . , k₆], and k_(i) is the stiffness (such as aspring constant) of each leg. From equations (23), (26), and (21),

F=J^(T)χJΔx=KΔx  (30)

This is a Hooke's Law relationship in which K is a stiffness factorthat, in this instance is a matrix (“stiffness matrix”) due to themultiple degrees of freedom of motion of the platform. The stiffnessmatrix is symmetric, positive semi-determinative, andconfiguration-dependent. The values of k_(i) desirably are equal toalleviate cross-coupling.

In equation (30), the Jacobian matrix is as expressed in equation (18),namely:

$J = \begin{bmatrix}s_{1}^{T} & \left( {b_{1} \times s_{1}} \right)^{T} \\\cdots & \cdots \\s_{6}^{T} & \left( {b_{6} \times s_{6}} \right)^{T}\end{bmatrix}$

(See also equation (24).) The unit stiffness can be obtained from:

K=J^(T)J.  (31)

The height, h, cannot be zero. The expression for h is as set forth inequation (10). Based on coordinate expressions as set forth earlierabove, the 6×6 Jacobian matrix can be computed as follows:

$J = \left\lbrack {\begin{matrix}\frac{c}{L} & \frac{- \left( {{- b} + \sqrt{a^{2} - c^{2}}} \right)}{L} \\\frac{- c}{L} & \frac{- \left( {{- b} + \sqrt{a^{2} - c^{2}}} \right)}{L} \\\frac{- \left( {{{- \sqrt{3}}b} + c + {\sqrt{3}\sqrt{a^{2} - c^{2}}}} \right)}{2L} & \frac{- \left( {b + {\sqrt{3}c} - \sqrt{a^{2} - c^{2}}} \right)}{2L} \\\frac{{\sqrt{3}b} + c - {\sqrt{3}\sqrt{a^{2} - c^{2}}}}{2L} & \frac{{- b} + {\sqrt{3}c} + \sqrt{a^{2} - c^{2}}}{2L} \\\frac{- \left( {{\sqrt{3}b} + c - {\sqrt{3}\sqrt{a^{2} - c^{2}}}} \right)}{2L} & \frac{{- b} + {\sqrt{3}c} + \sqrt{a^{2} - c^{2}}}{2L} \\\frac{{{- \sqrt{3}}b} + c + {\sqrt{3}\sqrt{a^{2} - c^{2}}}}{2L} & \frac{- \left( {b + {\sqrt{3}c} - \sqrt{a^{2} - c^{2}}} \right)}{2L}\end{matrix}\begin{matrix}\frac{\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)^{1/2}}{L} & \frac{{b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}}{L} \\\frac{\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)^{1/2}}{L} & \frac{{b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}}{L} \\\frac{\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)^{1/2}}{L} & \frac{- {b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}}{2L} \\\frac{\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)^{1/2}}{L} & \frac{- {b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}}{2L} \\\frac{\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)^{1/2}}{L} & \frac{- {b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}}{2L} \\\frac{\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)^{1/2}}{L} & \frac{- {b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}}{2L}\end{matrix}\begin{matrix}0 & \frac{- {bc}}{L} \\0 & \frac{bc}{L} \\\frac{{- \sqrt{3}}{b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}}{2L} & \frac{- {bc}}{L} \\\frac{{- \sqrt{3}}{b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}}{2L} & \frac{bc}{L} \\\frac{\sqrt{3}{b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}}{2L} & \frac{- {bc}}{L} \\\frac{\sqrt{3}{b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}}{2L} & \frac{bc}{L}\end{matrix}} \right\rbrack$

The 6×6 stiffness matrix K is as follows:

$K = {\quad\left\lbrack {\begin{matrix}\frac{{- 3}\left( {{- b^{2}} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}{L^{2}} & 0 \\0 & \frac{{- 3}\left( {{- b^{2}} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}{L^{2}} \\0 & 0 \\0 & \frac{{- 3}\left( {{- b} + \sqrt{a^{2} - c^{2}}} \right)}{L^{2}{b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}} \\\frac{3\left( {{- b} + \sqrt{a^{2} - c^{2}}} \right)}{L^{2}{b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}} & 0 \\0 & 0\end{matrix}\left. \quad {\begin{matrix}0 & 0 \\0 & 0 \\\frac{6\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}{L^{2}} & \frac{{- 3}\left( {{- b} + \sqrt{a^{2} - c^{2}}} \right)}{L^{2}{b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}} \\0 & \frac{3\; {b^{2}\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}}{L^{2}} \\0 & 0 \\0 & 0\end{matrix}\begin{matrix}\frac{3\left( {{- b} + \sqrt{a^{2} - c^{2}}} \right)}{L^{2}{b\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}^{1/2}} & 0 \\0 & 0 \\0 & 0 \\0 & 0 \\\frac{3\; {b^{2}\left( {L^{2} - b^{2} + {2b\sqrt{a^{2} - c^{2}}} - a^{2}} \right)}}{L^{2}} & 0 \\0 & \frac{6\; b^{2}c^{2}}{L^{2}}\end{matrix}} \right\rbrack} \right.}$

In the stiffness matrix, the upper-left 3×3 submatrix represents thetranslational (x, y, z) stiffness, the lower-right 3×3 submatrixrepresents the torsional (θ_(x), θ_(y), θ_(z)) stiffness, and the othersubmatrices represent cross-coupling effects between forces and moments,and between rotations and translations, respectively.

By way of example, consider a configuration in which:

Point x y A₁ −70 230.6 A₂ 70 230.6 A₃ 234.7 −54.7 A₄ 164.7 −175.9 A₅−164.7 −175.9 A₆ −234.7 −54.7 B₁₂ 0 170 B₃₄ 147.2 −85 B₅₆ −147.2 −85

a=[(230.6)²+(70)²]^(1/2)=240.99 mm

b=170.00 mm

c=70.00 mm

L=156.7 mm

$J = \begin{bmatrix}0.4467 & {- 0.3867} & 0.8068 & 137.1522 & 0 & {- 75.9413} \\{- 0.4467} & {- 0.3867} & 0.8068 & 137.1522 & 0 & 75.9413 \\{- 0.5583} & {- 0.1935} & 0.8068 & {- 68.5761} & {- 118.7773} & {- 75.9413} \\{- 0.1116} & 0.5802 & 0.8068 & {- 68.5761} & {- 118.7773} & 75.9413 \\0.1116 & 0.5802 & 0.8068 & {- 68.5761} & 118.7773 & {- 75.9413} \\0.5583 & {- 0.1935} & 0.8068 & {- 68.5761} & 118.7773 & 75.9413\end{bmatrix}$${{and}\mspace{14mu} K} = {\left( {1.0 \times 10^{4}} \right)\begin{bmatrix}0.0001 & 0 & 0 & 0 & 0.0159 & 0 \\0 & 0.0001 & 0 & {- 0.0159} & 0 & 0 \\0 & 0 & 0.0004 & 0 & 0 & 0 \\0 & {- 0.0159} & 0 & 5.6432 & 0 & 0 \\0.0159 & 0 & 0 & 0 & 5.6432 & 0 \\0 & 0 & 0 & 0 & 0 & 3.4602\end{bmatrix}}$

As noted above in equation (10), h²=L²−b²+2b√{square root over(a²−c²)}−a². If we define M=−b+√{square root over (a²−c²)} andN=a²+b²−2b√{square root over (a²−c²)}, then the stiffness matrix abovereduces to:

$K = \begin{bmatrix}\frac{3N}{L^{2}} & 0 & 0 & 0 & \frac{3{Mbh}}{L^{2}} & 0 \\0 & \frac{3N}{L^{2}} & 0 & \frac{{- 3}{Mbh}}{L^{2}} & 0 & 0 \\0 & 0 & \frac{6h^{2}}{L^{2}} & 0 & 0 & 0 \\0 & \frac{{- 3}{Mbh}}{L^{2}} & 0 & \frac{3b^{2}h^{2}}{L} & 0 & 0 \\\frac{3{Mbh}}{L^{2}} & 0 & 0 & 0 & \frac{3b^{2}h^{2}}{L^{2}} & 0 \\0 & 0 & 0 & 0 & 0 & \frac{3b^{2}c^{2}}{L^{2}}\end{bmatrix}$

Again, the upper-left 3×3 submatrix represents the translational (x, y,z) stiffness, the lower-right 3×3 submatrix represents the torsional(θ_(x), θ_(y), θ_(z)) stiffness, and the other submatrices representcross-coupling effects between forces and moments, and between rotationsand translations, respectively. Note that the cross-coupling terms havethe same magnitude, in this instance,

${\frac{3{Mbh}}{L^{2}}}.$

These terms become zero whenever M=0. In other words, there are nocross-coupling terms in the stiffness matrix if the followingrelationship is satisfied (see FIG. 3(B)):

a ² =b ² +c ²  (32)

Under such a situation, the stiffness matrix reduces further to:

$K = \begin{bmatrix}\frac{3c^{2}}{L^{2}} & 0 & 0 & 0 & 0 & 0 \\0 & \frac{3c^{2}}{L^{2}} & 0 & 0 & 0 & 0 \\0 & 0 & \frac{6\left( {L^{2} - c^{2}} \right)}{L^{2}} & 0 & 0 & 0 \\0 & 0 & 0 & \frac{3{b^{2}\left( {L^{2} - c^{2}} \right)}}{L^{2}} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{3{b^{2}\left( {L^{2} - c^{2}} \right)}}{L^{2}} & 0 \\0 & 0 & 0 & 0 & 0 & \frac{3b^{2}c^{2}}{L^{2}}\end{bmatrix}$

which is representative of a configuration in which the plane of eachpair of legs is perpendicular to the plane of the base. Note that, inthis instance, all the cross-coupling terms have been reduced to zero.

From the foregoing formulation of the stiffness matrix K, a change inleg interval c (or the included angle θ) will affect axial stiffness.For example, increasing c will increase x, y, and θ_(z) stiffness whiledecreasing z, θ_(x), and θ_(y) stiffness. Consequently, the leg intervalc is the design parameter that determines the stiffness of the platform.For equal stiffness at the translational-axis condition (x, y, and zaxes), the leg interval is:

$\begin{matrix}{c = {\sqrt{\frac{2}{3}}L}} & (33)\end{matrix}$

and the included angle (θ) (see FIG. 3(B)) of this embodiment issubstantially 109.5°. It is noted that equal stiffness is not arequirement for achieving zero cross-coupling. Rather, the key toachieving zero cross-coupling, as stated earlier above, is M=0. It isalso noted that the non-zero terms in the preceding matrix need notalways be equal to the respective values listed.

In an exemplary mechanical and control system for achieving 6 DOFmotion, two different coordinate-transformation matrices are used. Forconverting global-sensor location to top-mirror position, ageometric-coordinate transformation matrix T can be obtained by:

dŷ _(x) =T·dŷ _(s)  (34)

wherein the matrix T reflects a conversion of sensor-position data tomirror position. Also, the 6-DOF control output needs to be converted toeach individual leg command. The derived inverse of the Jacobian matrixwill suffice for this conversion. For small displacements, in an idealcase in which plant and sensor dynamics are ignored:

d{circumflex over (l)} _(q) =J·dŷ _(x)  (35)

A representative embodiment of the hexapod AMA mechanism is shown inFIGS. 4(A)-4(B). Turning first to FIG. 4(A), shown are a base 14 and aplatform 12, and an optical element (e.g., a mirror) 32 mounted to theunderside of the platform 12 by holds 34. The platform 12 is supportedrelative to the base by three pairs of legs L₁-L₂, L₃-L₄, and L₅-L₆, ofwhich legs L₁ and L₂ largely visible in the figure. The legs L₁-L₆ arenominally all of identical length, but each leg comprises a respectiveactuator (discussed later below) that produces, when actuated, a desiredamount of elongation of the respective leg. The discussion belowconcerning the legs L₁-L₂ is applicable to the other two pairs of legsL₃-L₄ and L₅-L₆.

A distal end of the leg L₁ is mounted to the base 14 by a block 30 a,and a distal end of the leg L₂ is mounted to the base by a block 30 b(also visible in FIG. 4(A) is the block 30 c by which a distal end ofthe leg L₃ is mounted to the base). At or near the respective distal endof each leg L₁-L₂ is a respective distal flexure 36 a, 36 b mounted tothe respective block 30 a, 30 b. The distal flexures 36 a, 36 b provideflexibility of the respective legs L₁, L₂ in the desired degrees offreedom (DOF) relative to the base 14 as required to accommodateextensions and retractions of the respective legs. For example, eachdistal flexure 36 a, 36 b is configured to provide a respective two DOFof motion (but not all six DOF) to the respective leg L₁, L₂ relative tothe base 14.

The proximal end of each leg L₁-L₂ has a respective proximal flexure 38a, 38 b (only the flexure 38 b is visible). The proximal flexures 38 a,38 b of the legs are connected to an apex block 40. The proximalflexures 38 a, 38 b provide flexibility of the respective legs L₁, L₂ inthe desired DOF relative to the platform 12 as required to accommodateextensions and retractions of the respective legs. For example, eachproximal flexure 38 a, 38 b is configured to provide at least two DOF ofmotion (but not all six DOF) to the respective leg L₁, L₂ relative tothe platform 12. The proximal flexures 38 a, 38 b work in coordinationwith the distal flexures 36 a, 36 b in this regard.

FIG. 4(A) shows particularly the legs L₁ and L₂ of this embodiment. Notethe splayed arrangement of the legs L₁, L₂, giving the pair an inverted“V” configuration in which the apex of the V corresponds to the proximalends conjoined at the apex block 40. The other two pairs of legs L₃, L₄and L₅, L₆ are similarly configured. A plan view of the embodiment isshown in FIG. 4(B), showing all the legs L₁-L₆ and blocks 30 a-30 f. Therespective apices of the three pairs of legs are situated equidistantlyon a circle on the platform 12 (see FIG. 3(A), showing the circle 16 onwhich the points B₁₂, B₃₄, B₅₆, corresponding to respective apices, arelocated). Note also that the distal ends of the legs (at the respectiveblocks 30 a-30 f) are situated on a circle on the base 14 (see FIG. 1,showing the circle 18 on which the points A₁-A₆, corresponding torespective distal ends, are located).

Each leg L₁-L₆ includes a respective length monitor 42 a-42 f thatmeasures and monitors the length (including changes in length) of therespective leg. The leg-length monitors 42 a-42 f can be highly accurateencoders utilizing, for example, laser scales. The leg-length monitors42 a-42 f are especially advantageous when used in feedback-controlsystems for controlling extension and retraction of the legs, includingin real time.

To effect changes in their length, each leg L₁-L₆ comprises a respectivepiezoelectric (e.g., PZT-based piezo-ceramic) actuator 44 a-44 f. InFIG. 5 only one piezoelectric actuator 44 a can be seen. As previouslynoted, all the legs L₁-L₆ have nominally the same length. Thepiezoelectric actuators 44 a-44 f impart very small changes in length tothe respective legs L₁-L₆ as required to perform a fine positionaladjustment of the platform 12 (and optical element mounted on it)relative to the base 14. For example, in this particular embodiment, therange of leg-length change achievable by the actuators is tens ofmicrometers. Leg extension (increasing the length of the leg) isachieved by energizing the respective actuator sufficiently to cause adesired increase in length of the actuator. Leg retraction (decreasingthe length of the leg) is achieved by reducing the degree ofenergization of the respective actuator sufficiently to cause a desiredreduction in length of the actuator.

For measuring height (and changes in height) of the platform 12 relativeto the base 14 (or other fixed reference) accompanying a particularextension or retraction of one or more of the legs, each pair of legsL₁-L₂, L₃-L₄, L₅-L₆ has an associated height monitor 46 a, 46 b, 46 cmounted to the base 14. The height monitors 46 a-46 c measure heightalong a respective line that is perpendicular to the base and thatpasses through the apex of the respective pair of legs. Additionalpositional monitoring is performed by monitors 48 a, 48 b, 48 c situatedmidway between pairs of legs and mounted to the base 14. The heightmonitors 46 a-46 c and the position monitors 48 a-48 c can be highlyaccurate encoders or interferometers that measure displacement relativeto a stationary frame of reference such as a lens barrel holding theoptical elements of the system. If an encoder is used, it can be basedupon reflection or transmission of light.

Whereas the base 14 can be used as a positional reference for themonitors, use of the base as a reference may not be practical in certaininstances. For example, movements of the optical element 32 relative tothe base 14 can generate vibrations in the base, thereby creating amoving reference.

Turning now to FIG. 5, the legs L₁ and L₂ are shown, including theblocks 30 a, 30 b; the distal flexures 36 a, 36 b, the proximal flexures38 a, 38 b, and the apex block 40. Note that each leg L₁, L₂ has arespective leg axis Ax₁, Ax₂, and that the angle θ between the leg axesis as noted in FIG. 3(B). The axes Ax₁, Ax₂ collectively define a “legplane” that, when the legs L₁, L₂ are mounted between the base 14 andplatform 12, is perpendicular to the plane of the base 14. Thisperpendicularity of the leg planes is apparent in the view of FIG. 4(B),which depicts the leg plane P₁ for the legs L₁, L₂, the leg plane P₂ forthe legs L₃, L₄, and the leg plane P₃ for the legs L₅, L₆. The existenceof these leg planes P₁-P₃ relies upon satisfaction of the relationshipa²=b²+c², as noted in equation (32), wherein a, b, and c are as shown inFIG. 3(A).

Returning to FIG. 5, and using the leg L₁ as an example (and referringto components associated with the leg L₁ as exemplary of correspondingcomponents on the other legs), each leg has a respective piezoelectricactuator 44 a, which exhibits extension when the actuator iselectrically energized. The piezoelectric actuator 44 a includes afine-motion actuator 50 a and a coarse-motion actuator 50 b. In thisembodiment the fine-motion actuator 50 a comprises one piezoelectricelement, and the coarse-motion actuator 50 b comprises multiplepiezoelectric elements arranged in tandem along the leg axis Ax₁. Thefine-motion actuator 50 a and coarse-motion actuator 50 b also arearranged in tandem along the leg axis Ax₁. Each of the actuators 50 a,50 b is separately energized by a respective driver (not shown). Thepiezoelectric actuator 44 a is situated in a yoke 52 that allows thepiezoelectric actuator 44 a to apply an extension force (one DOF ofmotion) strictly along the longitudinal axis A₁ of the respective legL₁. Compliance along the axes A₁, A₂ of the legs L₁, L₂ to accommodateleg extensions is provided by respective flexures 56 a, 56 b.

By way of example, the coarse-motion actuator 50 b is configured toprovide an accuracy of actuation performance in the micrometer range,and the fine-motion actuator 50 a is configured to provide an accuracyof actuation performance in the nanometer range. For some applications,one of the portions (the fine-motion actuator 50 a) of the piezoelectricactuator can be omitted if the particular application does not requireit.

The optical system with which a hexapod as described above can beassociated can be any of various reflective, catadioptric, refractive,and other types of optical systems including combinations of thesespecific systems. In general, the optical system can be any such systemthat is used under conditions requiring adjustability in the nm range aswell as 6 DOF of movement. An example system is shown in FIG. 6, inwhich the system includes six mirrors PM1-PM6, as exemplary opticalelements of the system, all mounted to a “frame” F (e.g., an optical“barrel” or “column”). The depicted system is particularly suitable foruse as a projection-optical system for performing EUV microlithography.In the depicted system the second mirror PM2 is mounted on a hexapodmounting as described above.

Whereas optical-element mountings have been described above in thecontext of representative embodiments, it will be understood that thesubject mountings are not limited to those representative embodiments.On the contrary, the subject optical-element mountings are intended toencompass all modifications, alternatives, and equivalents as may beincluded within the spirit and scope of the following claims.

1. A hexapod kinematic mounting, comprising: a base defining a baseplane; a platform situated relative to the base and movable relative tothe base; and six legs each having nominally identical length and arespective leg axis, the legs having substantially equal stiffness andbeing arranged in three pairs of legs extending between the base andplatform and supporting the platform relative to the base, each pair oflegs having first and second ends, the first ends of each pair beingcoupled together in a Λ-shaped manner forming a respective apex and therespective second ends being splayed relative to the apex, the apicesbeing situated substantially equidistantly from each other at respectivelocations on a circle on the platform, and the respective second ends ofthe pairs of legs being mounted at respective locations on a circle onthe base such that the respective axes of each pair of legs define arespective leg plane that is substantially perpendicular to the baseplane, each leg comprising an actuator serving, when energized, tochange a length of the respective leg such that a coordinatedenergization of the respective actuators in selected legs produces adesired movement of the platform relative to the base in all six degreesof freedom of motion.
 2. The mounting of claim 1, wherein the respectivelegs of each pair form an angle of substantially 109.5° at therespective apex.
 3. The mounting of claim 1, wherein the leg actuatorsare respective piezoelectric actuators.
 4. The mounting of claim 3,wherein each piezoelectric actuator comprises a respective coarseactuator and a respective fine actuator.
 5. The mounting of claim 4,wherein the coarse actuator and the fine actuator are arranged in tandemalong the respective leg axis.
 6. The mounting of claim 1, wherein eachleg further comprises a respective leg-extension flexure situatedrelative to the leg actuator to provide at least one, but not all six,degrees of freedom of motion accompanying leg extension and retractioncaused by the respective leg actuator.
 7. The mounting of claim 1,wherein each leg further comprises a respective leg-length monitor. 8.The mounting of claim 1, wherein: the first ends of each pair of legscomprise respective flexures providing the respective end with at leasttwo, but not all six, degrees of freedom of motion; and the second endsof each pair of legs comprise respective flexures providing therespective end with at least two, but not all six, degrees of freedom ofmotion.
 9. The mounting of claim 1, further comprising at least oneheight monitor situated and configured to measure and monitor positionof the platform relative to a fixed reference.
 10. The mounting of claim9, wherein the fixed reference is the base.
 11. A kinematically mountedoptical element, comprising: an optical element; a base defining a baseplane; a platform movable relative to the base; at least one holdaffixing the optical element to the platform; and a hexapod situatedbetween the base and the platform so as to support the platform relativeto the base, the hexapod comprising six legs each having a respectiveleg axis, the legs having nominally identical length and substantiallyequal stiffness and being arranged in three pairs each having first andsecond ends, the first ends of each pair being coupled together in aΛ-shaped manner forming a respective apex and the respective second endsbeing splayed relative to the apex, the apices being situatedsubstantially equidistantly from each other at respective locations on acircle on the platform, and the respective second ends of the pairs oflegs being mounted at respective locations on a circle on the base suchthat the respective axes of each pair of legs define a respective legplane that is substantially perpendicular to the base plane, each legcomprising an actuator serving, when energized, to change a length ofthe respective leg such that a coordinated energization of therespective actuators in selected legs produces a desired movement of theplatform relative to the base in all six degrees of freedom of motion.12. The optical element of claim 11, wherein the optical element is amirror.
 13. The optical element of claim 11, wherein the respective legsof each pair form an angle of substantially 109.5° at the apex.
 14. Anoptical system, comprising: a frame; a base mounted to the frame anddefining a base plane; a platform movable relative to the base; anoptical element mounted to the platform; and a hexapod situated betweenthe base and the platform so as to support the platform relative to thebase, the hexapod comprising six legs each having a respective leg axis,the legs having nominally identical length and substantially equalstiffness and being arranged in three pairs each having first and secondends, the first ends of each pair being coupled together in a Λ-shapedmanner forming a respective apex and the respective second ends beingsplayed relative to the apex, the apices being mounted equidistantlyfrom each other at respective locations on a circle on the platform, andthe respective second ends of the pairs of legs being mounted atrespective locations on a circle on the base such that the respectiveaxes of each pair of legs define a respective leg plane that issubstantially perpendicular to the base plane, each leg comprising anactuator serving, when energized, to change a length of the respectiveleg such that a coordinated energization of the respective actuators inselected legs produces a desired movement of the platform relative tothe base in all six degrees of freedom of motion.
 15. The optical systemof claim 14, wherein the respective legs of each pair form an angle ofsubstantially 109.5° at the apex.
 16. The optical system of claim 14,wherein the leg actuators are respective piezoelectric actuators. 17.The optical system of claim 16, wherein each piezoelectric actuatorcomprises a respective coarse actuator and a respective fine actuator.18. The optical system of claim 14, wherein each leg further comprises arespective leg-extension flexure situated relative to the leg actuatorto provide at least one, but not all six, DOF of motion accompanying legextension and retraction caused by the respective leg actuator.
 19. Theoptical system of claim 14, wherein each leg further comprises arespective leg-length monitor.
 20. The optical system of claim 14,wherein: the first ends of each pair of legs comprise respectiveflexures providing the respective end with at least two, but not allsix, DOF of motion; and the second ends of each pair of legs compriserespective flexures providing the respective end with at least two, butnot all six, DOF of motion.
 21. The optical system of claim 14, furthercomprising at least one monitor situated and configured to measure andmonitor position of the optical element relative to a fixed reference.22. The optical system of claim 14, wherein the optical element is areflective optical element.
 23. The optical system of claim 14, whereinthe optical system is an EUVL optical system.
 24. The optical system ofclaim 23, wherein the optical system is an EUVL projection-opticalsystem.
 25. The optical system of claim 14, wherein: the optical systemcomprises multiple optical elements; and at least one optical element ismounted to the frame by a respective base, platform, and hexapod.
 26. Akinematically mounted optical element, comprising: an optical element; abase defining a base plane; a platform movable relative to the base; atleast one hold affixing the optical element to the platform; and ahexapod situated between the base and the platform so as to support theplatform relative to the base, the hexapod comprising six legs eachhaving a respective leg axis, the legs having nominally identical lengthand being arranged in three pairs each having first and second ends, thefirst ends of each pair being coupled together in a Λ-shaped mannerforming a respective apex and the respective second ends being splayedrelative to the apex, the apices being mounted equidistantly from eachother at respective locations on a circle on the platform, and therespective second ends of the pairs of legs being mounted at respectivelocations on a circle on the base such that the respective axes of eachpair of legs define a respective leg plane that is substantiallyperpendicular to the base plane, each leg comprising an actuatorserving, when energized, to change a length of the respective leg suchthat a coordinated energization of the respective actuators in selectedlegs produces a desired movement of the platform, with substantially nocross-coupling, relative to the base in all six degrees of freedom ofmotion.
 27. The optical system of claim 26, wherein the legs havesubstantially equal stiffness.
 28. The optical system of claim 26,wherein the respective legs of each pair form an angle of substantially109.5° at the apex.
 29. An optical system, comprising an optical elementas recited in claim
 26. 30. A hexapod kinematic mounting, comprising: abase defining a base plane; a platform situated relative to the base andmovable relative to the base; and six legs each having nominallyidentical length and a respective leg axis, the legs being arranged inthree pairs of legs extending between the base and platform andsupporting the platform relative to the base, each pair of legs havingfirst and second ends, the first ends of each pair being coupledtogether in a Λ-shaped manner forming a respective apex and therespective second ends being splayed relative to the apex such that therespective legs of the pair form an angle of substantially 109.5° at theapex, the apices being mounted equidistantly from each other atrespective locations on a circle on the platform, and the respectivesecond ends of the pairs of legs being mounted at respective locationson a circle on the base such that the respective axes of each pair oflegs define a respective leg plane that is substantially perpendicularto the base plane, each leg comprising an actuator serving, whenenergized, to change a length of the respective leg such that acoordinated energization of the respective actuators in selected legsproduces a desired movement of the platform relative to the base in allsix degrees of freedom of motion.